The 32-year-old Serbia central defender said last month he would be leaving the Premier League champions at the end of this term and was looking at moving to a team outside England.
"Vidic is a world-class player," Inter president Erick Thohir told the club's website (www.inter.it).
"I'm extremely happy to have completed this deal to bring Nemanja to Milan. He's one of the world's best defenders and his qualities, international pedigree and charisma will be an asset to the team and help bring through our younger players.
"He will bring added value to the club and is another piece of the great Inter side we are building,"
Vidic joined United in 2006, winning a host of trophies, including five Premier League titles, the Champions League and the Club World Cup.
(Reporting by Josh Reich; Editing by Ken Ferris)